Normals of mesh are super small like a dot, what could have possibly gone wrong?
Sounds like your object may have some unapplied object scale.
Press &#9096; CtrlA> Scale in object mode to apply it.

The length of the normal indicators is defined in object space; scaling the mesh down in edit mode makes the indicators "longer" relative to the mesh:



But scaling in object mode scales object space itself, keeping the indicator size proportional to the mesh size. So if you have a huge object scaled down a lot, you'll end up with very small indicators.

Object scale behaves like a sort of offset, only affecting the apparent size of the object while mesh operators such and "see" the object without any object scaling. This can affect other things besides normal indicators, so it's a good idea to be aware of this behavior.

See Why are the longer edges being beveled more?